What is the difference between the ant system and the max-min ant system?
Let's start by making clear that both AS and MMAS use only global pheromone update. Now, the MMAS has two main differences regarding AS:

In AS, all ants that completed a solution are used for the update, while in MMAS only the best ant with a complete solution is used for update (as you had pointed out).

In AS, the pheromone values are not explicitly bounded. In MMAS, the pheromones are enforced to lie within a pre-set interval $\tau_{min} \leq \tau_{ij} \leq \tau_{max}$ (which gives its name to the algorithm). To ensure this condition, the pheromone update is done by means of the formula


$$
\tau_{ij} \leftarrow \left[ (1-\rho) \cdot \tau_{ij} + \Delta \tau_{ij}^{best} \right]_{\tau_{min}}^{\tau_{max}},
$$
with the operator $[x]_a^b$ defined as
$$
[x]_a^b= 
\begin{cases}
    a \qquad \mathrm{if} \quad x &gt; a\\
    b \qquad \mathrm{if} \quad x &lt; b\\
    x \qquad \mathrm{otherwise}
\end{cases}
$$
Reference: Ant Colony Optimization. Artificial Ants as a Computational Intelligence Technique.